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department
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Steady State Unemployment Under Profit Sharing
Martin
No.
399
L.
Weitzman
November 1985
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
4
1986
Steady State Unemployment Under Profit Sharing
Martin
No.
399
L.
Weitzman
November 1985
DECS
1986
Martin L. Weitzman
October 30, 1985
"Steady State Unemployment Under Profit Sharing"
Introduction
This paper has several aims.
First of all, it seeks to tell a reaon-
able story about how asymmetric treatment of high-seniority "insider" workers
and non-tenured "outsiders" can give rise to bad macroeconomic steady states
in a wage economy.
The ultimate source of the unemployment-inflation dilemma
in this story is the idea that low-seniority outsider workers end up unem-
ployed because they have too little voice in negotiations over wages.
The
paper then considers profit sharing as a possible alternative payment mechanism having the automatically corrective incentive property that employers
always want to hire more outsiders.
Given the assumptions of the model, it
is shown that widespread profit sharing will result in lower unemployment and
more output even though it is individually rational for insiders always to
prefer wages over profit shares.
Put more fancifully, a wage system has a
negative macroeconomic externality, while a profit-sharing system has favorable externality effects on employment and, indirectly, on price stability.
If the logic of the externality argument is accepted the path is open for
government policy to encourage widespread profit sharing as an instrument for
lowering the iNAIEU.
I am very grateful to Dennis Snower and an anonymous referee for their
unusually helpful and thorough comments on an earlier draft.
They forced me
to rethink through what I was doing, and I hope the paper is the better for
it.
My colleague Robert M. Solow also made useful suggestions.
None of
these people should be blamed for the inadequacies that remain nor do they
necessarily agree with my conclusions.
Support from the National Science
Foundation is gratefully acknowledged.
The Cast of Characters
There are four key players in the models
insider workers, outsider
workers, firms, and the government.
In making much hinge on the distinction between insider workers and
outsider workers,
I
am following the pioneering work of Lindbeck and Snower.^
They identify insiders as those who already have a job and outsiders as those
who are unemployed or laid off.
A theory or paradigm is then developed,
based largely on the asymmetric cost of replacing insiders by outsiders,
which provides a microeconomic explanation for the prevalence of involuntary
unemployment.
The theory clarifies why outsiders do not underbid insiders,
posits a major role for unions, and suggests novel policy measures to lower
unemployment by reducing the market power of insiders vis-a-vis outsiders.
Overall,
I
believe the insider-outsider paradigm has a strong resonance of
plausibility and provides the most reasonable microeconomic framework yet put
forth for thinking about unemplojrment-related issues.
Actually,
I
would perhaps go even further than Lindbeck and Snower in
stressing the deep-seated nature of the insider-outsider paradigm, which in
my opinion transcends mere job markets and labor unions.
In every sphere of
social relations there tends to be a fundamental distinction between the
relatively sympathetic treatment of those insider members of the family,
community, or nation with whom we identify, and more of a hands-off attitude
toward those nameless, faceless outsiders with whom we do not identify.
Indeed, much political and social conflict is, in essence, about where to
draw this line, or at least where to shade a grey area.
The economic theor-
ist's highly abstract possibility of "side payments" between employed and
unemployed workers is only one of many factors in labor relations.
If in-
siders are treated so differently from outsiders in the world at large, and
so much hinges on the distinction,
why should the workplace be entirely ex-
empt from such influences?
In any event,
I
will here define insiders as workers having high
seniority and outsiders as those having low seniority.
A key assumption is
that pay bargains are negotiated between insider workers and firms, with
outsiders having very little say.
This might be envisioned as the ultimate
outcome of a bargaining process between management and a majority-rule labor
union where layoffs are by seniority
—
in which case the union's preferences
are those of the steady-state median voter.
Or,
it might come about more
generally for the kinds of reasons previously alluded to.
In either case it
will be convenient to maintain the fiction that long run equilibrium
bargaining takes place between a firm that is essentially interested in profits and a representative tenured worker essentially interested in pay.
There is a second basic assumption that may also be viewed as contro-
versial.
I
will assume that, at least in the long run, the firm is on its
Once a pay contract has been signed, the union
demand for labor schedule. 2
cannot restrict the company from eventually hiring as many workers as it
wants on the given contract.
Labor-management negotiations are essentially
limited to bargaining over pay parameters, while hiring of new workers is, in
labor-relations parlance, "not a mandatory subject to bargaining."
Granted
that this will sometimes result in ex-post inefficient pay-employment combi-
nations,
^
and whether or not it can ultimately be "explained" in terms of
transactions costs, enforcement technologies, informational asymmetries, or
other buzzwords of contemporary economic theory, the idea that the hiring
decision is essentially a management prerogative seems so deep-seated a
feature of capitalism that
I
believe it may fairly be accepted as a useful
point of departure for the purposes of this paper.
The Basic Idea
The principle on which this paper rests is quite simple.
firm controls the employment decision.
a
Suppose the
Then, other things being equal, under
profit-sharing system the firm is more inclined to expand employment and
output than under the corresponding wage system.
The following example illustrates the basic point.
Suppose the typ-
ical firm can produce output Y out of labor L by the production function
F(L).
Let the firm's revenue function be R(Y).
If the firm pays a fixed
wage w it will hire labor and produce output at the level where profits
R(F(L)) - wL are being maximized, or where the marginal revenue product of
labor R'F' equals the wage w.
Now imagine, as a kind of thought experiment, that a profit-sharing
system is put in place promising to pay each worker a base wage
X of gross profits per capita.
The firm's net profits are now
Each worker is now paid W =
(
1
-\) (R(F(L)
value of an extra worker is (1-x)(R'F' -
u))*
)
- ojL),
co
+
to
and a share
xi——
)
and the net marginal
Provided only that
oj<w,
the
firm will wish to expand output and employment from its previous position.
No matter how one interprets the "other things being equal," a profit-sharing
system is more expansionary than a wage system.
If pay parameters are set so
that workers are initially paid the same amount immediately after conversion
from a wage into a profit-sharing system, the firm will wish to expand em-
ployment and output, thereby contracting its price (and pay).
If pay para-
meters are set so that, after the firm's reaction to the introduction of a
profit-sharing contract, each worker ends up being paid the same as under the
previous wage contract, then output and employiQent must be higher, with
prices lower.
wage
CO,
The expansionary effects are stronger the smaller is the base
irrespective of the profit-sharing coefficient \.
There is a simple explanation of all this.
When factor costs are
lowered, a profit-maximizing monopolist will want to hire more input, produce
more output, and charge a lower price.
on the
When faced with a pure profits tax,
other hand, the monopolist will choose to hire the same amount of
input, produce the same output, and charge the same price.
So far as the
monopolist is concerned, conversion from a wage system to a profit-sharing
system (with a smaller base wage) is equivalent to lower factor costs coupled
with a pure profits tax.
Hence the expansionary bias of a profit sharing
system over a wage system.
This suggests that near-universal profit sharing might create full
emplojnnent,
or something close to it, with corresponding strong implications
for macroeconomic theory and policy.
These themes have been explored in
models where the labor market is essentially competitive
—
although slug-
gish,
so that pay parameters are the slowest adjusting variables in the sys-
tem.'*
The standard result is that in long run equilibrium, wage and profit-
sharing systems are isomorphic.
But a profit-sharing system displays "ex-
cess demand for labor" in the sense that firms would like to hire more labor
than they are actually able to hire on the equilibrium profit-sharing contract parameters they themselves have chosen.
With sticky pay parameters,
this means that a profit-sharing system tends to preserve full employment
even out of equilibrium, whereas a contractionary shock to a wage system is
more likely to cause unemployment.
It was argued
heuristically that this
basic contrast between wage and profit-sharing systems should carry over to
situations where labor has more than atomistic market power
—
where,
for
example, pay parameters have been pushed up by some fraction of their compet-
itive equilibrium values.
In such situations,
it was claimed,
realpolitik
wage economies are more likely to end up in regimes with greater unemployment
than their realpolitik profit-sharing analogues. 5
The present paper attempts
to tackle more formally the problem of specifying a long run resting point
(around which the short run fluctuations occur) when insider labor has non-
negligible market power.
The Model
Consider an economy consisting of a large fixed number^ of symmetric
monopolistically competitive firms.
Each firm produces output from labor
according to the production function
Y = F(L)
(1)
The firms are identical except that each is the sole producer of a
slightly, but sjonmetrically, differentiated product.
If one firm is charging
a price P while all the other firms are charging a price P,
demand for the
one firm's product will be
Y
=
D(P/P;M/P)
(2)
In the above demand function, M is a quasi-fixed shift variable stand-
ing for various aggregate demand factors.
Throughout the paper M will be
treated as an exogenously given macroeconomic parameter, indirectly dependent
on various policy variables like government spending, the money supply,
taxes, and so forth, whose exact specification is not essential to the argu-
ment.
More specifically, and without additional loss of generality, let M
simply represent, in reduced form, total economy-wide money expenditures
divided by the number of firms.
Demand for one firm's product can be written
in the particular form (2) without any loss of generality because of the
requirement that the firm's demand fiinction should be homogeneous of degree
zero in P, P, and M.
To make sense in a symmetric situation,
ftmction (2) should have the
M/P
=
the demand
property
D(1;M/P)
(3)
From (2), the firm's revenue function is
R(Y;M,P) = Y P D~''(Y;M/P)
(4)
(This paper follows the notational convention that the independent
variable to the left of the semi-colon symbol ";" represents the primary
functional relationship.
Variables to the right of the semi-colon are to be
viewed more like subscripted quasi-fixed parameters.
Inverses and deriva-
tives, when not otherwise specified, are taken with respect to the primary
independent variable.
Sometimes variables to the right of the semi-colon
will be suppressed for notational convenience in an unambiguous context.)
Suppose that a firm pays its workers by the profit sharing formula
W(.,X,L,M,?)
=
In the above formula
R(F(L);M,?) - .L
. . ,
L
to
__
^^^_^^
,
^R(P(L)i
^^^
Jj
represents the base wage, while X stands for the
share of gross profits per capita paid out to each worker.
Both
oj
and X are
treated as quasi-fixed contractual parameters in the short run, while the
firm is relatively free to adjust
L.
The firm's profits are then
n(w,X,L,M,P)
=
R(F(L);M,P) -
V(a) ,X
,L,M,P). L
(6)
Using (5), equation (6) can be rewritten as
n(a),X,L,M,P) = (l-x)(R(F(L);M,P) -
u)L)
(?)
Long Run Equilibrium
Suppose that the insider workers permanently attached to each firm are
organized into a union having some bargaining power.
Layoffs are strictly by
seniority, and the union members vote on any compensation package using the
principle of majority rule.
Then,
in any long run steady state equilibrium,
the union's preferences are those of the median-tenure voter who,
on the
margin, will always favor a pay raise over increased employment (so long as
it does not threaten his job,
plant closings).
e.g., by forcing mass layoffs via large scale
The counter force to ever-higher pay demands from the side
of the union is greater resistance from the side of a management increasingly
undesirous of granting them.
paper would
The basic qualitative results of the present
also hold when the union puts some weight on employment, so long
as it is small relative to the weight put on pay.
empirical generalization, and
I
Even if it were not a fair
believe it is, the idea that the union cares
a lot more about insider pay than about outsider jobs might still be viewed
as a theoretically interesting polar assumption because without it the model
is incapable of generating steady state unemployment.
The union wants high pay ¥, while management wants high profits
These objectives are in conflict.
IT-
The resolution is taken to be a weighted
Nash bargaining solution which maximizes, for each firm, the function
¥^n^-^
(8)
where b is a parameter representing the relative bargaining strength of the
union compared to the firm.
Note that it wou±d make no difference for the
Nash bargaining solution if, instead of being expressed in nominal pay and
profits, the formula (8) were expressed in terms of real pay W/P, or relative
pay V/¥, or real profits n/P, since one firm and its union have no control
over
the.
price level or pay elsewhere.
In effect,
the outcome of the bargaining process can be described as
follows.
Given any base wage and profit-sharing coefficient, by hypothesis
the firm will set the employment level to maximize profits,
mines (in a profit-sharing system) the level of pay.
which also deter-
The "Nashian arbitra-
tor" sets pay parameters at those levels which maximize
(8)
for each firm,
taking account of the Stackelberg-like profit-maximizing response of the
firm.
Some justification for maximizing a function of the form (8) can be
found in recent work on the theory of repeated games.
The threat point in
the repeated game here is the strike or lockout, which by hypothesis yields
W=0, n=0, while b in this context stands for possible asymmetries in the
bargaining procedure, and in the parties' beliefs.^
The essential logic of
the paper is unaltered when there are different threat point values,
although
the algebra is made considerably more complicated.
The question of why,
if there is involuntary unemployment,
the firm
does not negotiate with outsider workers, who are willing to do the insiders'
work for lower pay, is not directly addressed here.
Some answers in terms of
hiring-firing costs, cooperation harassment activities, disruptive behavior,
and the like are given in Lindbeck and Snower [1984].
Answers in terms of
the opportunity cost of time spent in bargaining are given in Shaked and
Sutton [1984].
to.
There may even be "sociological" reasons, previously alluded
In this paper
I
merely assume as a polar case that it simply is not
feasible for the firm to replace tenured insider workers by non-tenured outsiders, and hence, to a first approximation, relative bargaining strengths
are independent of the outsiders.
It is slightly more convenient to work with a logarithmic version of
(8)
—
the function
V = b log(W)
+
(1-b)
log(n)
(9)
10
Suppose, as a kind of thought experiment, that the profit-sharing coef-
ficient \ is statutorily fixed by the government at some predetermined value,
—
(it is not clear how this should be done in practice
tives on profit-sharing income.)
perhaps by tax incen-
The base wage will then be negotiated to
maximize (9). allowing for the firm's profit-maximizing employment reaction.
Without loss of generality, let labor units be normalized so that the
economy-wide full employment level divided by the number of firms is one.
Then the unemplojnnent rate u is related to the average employment level per
firm L by the simple formula
=
u
1
-
L
(10)
Suppose that all firms except one have negotiated a base wage
then selected an employment level L (<1) and price P.
co
»
and
Let the one firm's
profit maximizing amount of labor to hire as a function of the relevant
variables be formally denoted
L(co;w,L,\,M,P)
(11)
The above employment function,
sometimes denoted
L(to)
for brevity, must
satisfy the following conditions:
Case
I
(labor is demand constrained): if L<1,
from (ll) solves
L(a))
n(co,\,L(u)),M,P) = max n(u),\,L,M,P)
(12)
L>0
(¥ith unemployed labor available, the profit-maximizing firm will hire
workers to the point where the marginal revenue product of one more equals
the base wage
.
Case II (labor is supply constrained): if L=1
L(a})
,
from (11) solves
n(a),\,L(a,),M,P) = max n(co,X,L,M,P)
L
subject to:
co
+
\(R(F(L) )-cjL)/L
(13)
_
>
^
+ X
(
PF (
)
1
-[^J")
(When the labor force is fully employed, the firm is constrained to pay
11
at least the going rate.
Substituting
L(ci))
from (11) into (9), the logarithm of the weighted
Nash bargaining function with built-in employment reaction becomes
V(a);a),L,\tMrP)
b log
=
W(wA,L(w),M,P)
+
(1-b)
log n(w,\,LU),M,P)
(14)
The following concept of a long run equilibrium will be used.
Definition:
A symmetric Nash equilibrium is a triple (a)*,L* (<l),P*)
simultaneously satisfying:
V(cj*;a)*,L*,\,M,P*) = max
(i)
V(tj
;w*,L*,\ ,M,P*)
(15)
(jO
(o)*
maximizes the weighted Nash product of pay and profits.)
(ii)
L(a,*;co*,L*,\,M,P*) = L*
(16)
(L* is the profit maximizing value of L.)
(iii)
P* = R(F(L*);M,P*)
(17)
(P* is the profit maximizing value of P.)
(Note that the macroeconomic equilibrium condition
P*.F(L*) = M
(18)
is then automatically guaranteed by (3)«)
The Pure Wage System
At this point it is useful to treat specially the case X=0,
both to
gain an intuitive feel for the long run equilibrium concept (15)-(17) in a
familiar setting, and also to motivate an additional condition that must be
imposed on the model economy for solutions to make sense.
Much of the rea-
soning for the simpler case X=0 can be carried over to the case X>0.
Suppose, then, that the government has statutorily fixed X at zero.
Profit sharing is thereby forbidden, and the economy is left with a pure wage
12
system.
of
To emphasize the point,
let the wage here be denoted w (instead
U)).
The long run equilibrium (15)-(17) can be solved as follows.
Suppose
first the wage economy is in a regime with positive unemployment:
u = 1-L*
0.
>
(19)
Then, suppressing extra variables where the meaning is otherwise clear,
each firm's profit function is
n(w) = R(F(L(w))) - wL(w),
(20)
R'(F(L(w))).F'(L(w))
(21)
where
=
w
With unemployed labor available to be hired (condition (19)), the Nashian arbitrator's maximand
V(w) = b log(w) + (1-b) log(n(w))
(22)
will have an interior solution satisfying
V'(w)
=
(23)
0,
or
^.
i^-l)Kkl
w
n(w)
(24)
Since from duality theory
n'(w) = -L,
(25)
condition (24) can be rewritten as
1
=
n(w)
b
s
^26)
c
In any symmetric equilibrium situation where every firm is hiring the
same amount of labor L and charging the same price
P,
the price elasticity of
demand faced by the firm is, from (3),
,(L)=^1U^
(27)
The monopoly "markup coefficient" of price over marginal cost is then
13
p(L) .
^
(28)
SO that the profit maximizing price is
P = w ^(L)/F"(L)
(29)
Profits are then
n
= PF(L)
-
wL
=
w.
(^(L)F(L)/F'(L)
-
L)
(30)
Substituting (30) into (26) yields the basic condition
1
=
(
^
U(L*) F(L*)
L* F'(L*)
^
^
_
Jb_
^^^)
^
1-b
for the optimal solution L*(b).
Equation
(31 )»
representing a long run equilibrium condition, is, of
course, expressed in purely "real" terms.
Nominal variables are tacked on to
the long run equilibrium condition (31) by equations
(18) and
(29).
Thus,
in
the long run, P and w are proportional to M.
Note that any attempt to use macroeconomic policy to stimulate the
economy toward a higher level of employment than L* by, in effect, increasing
M,
might temporarily succeed in a short run when wages are quasi-fixed.
But
employment would decline back to L* in the longer run when wages are renegotiated up, and that would be passed on into a correspondingly higher price
level.
In this sense
u(b) =
1
- L*(b)
(32)
represents a NAIEU-like "natural-rate" unemployment level in the present
model.
The expression
S(L) .
4%f^
-
1
(33)
that appears in (31) is the ratio of profits to the wage bill, or "capital's
share" divided by "labor's share", expressed as a function of the employment
14
level L=1-u.
It
S'(L)
is assumed that S(L) is pro-cyclical,
or that
(M)
>
As well as making good economic sense in its own right,
in equation (31
is needed to make L*(b)
(33),
behave sensibly.
)
condition (34)
Using definition
let b(0) be defined as the full employment cutoff value of b that sat-
isfies (31) for L*=1
1-S(l)-«°i^
(35)
in the above equation can be explicitly solved out to
The variable b(0)
yield
(3l)-(36) may conveniently be sum-
The economic content of conditions
There exists a "cutoff value" b(0) of
marized by the following implications.
b,
0<b(0)<1, such that b<b(0) implies full employment, while b>b(0) means u
Furthermore, in the range b>b(0),
(the unemployment rate) is positive.
u'(b)
(37)
>
The condition (37)
>
which should hold for the model to make sense,
might be assumed initially and then turned around to imply (34), if it were
felt there were not already sufficient economic justification for (34)-
Using definition (33) to rewrite
V(w) -l(b
^;!7\
^
V
as
J
(38)
S(L(w;j
w
it is routine to establish, using
(34),
that V(w) is a quasi-concave
function.
Note that, under widespread unionization, insider workers may not actu-
ally benefit much from an economy-wide increased value of b, if they do at
all.
The real wage w/P
=
F'(L)/^i(L)
(from (29)) is probably not strongly
15
related to employment levels
L.
The economy-wide effect of universally
increased insider bargaining power is primarily to cause greater unemployment
among outsiders, from (37), not to increase the real wages of the insidersCombining
(
18)
,
(29)
w'(b)
,
(54)
,
(37)
,
it is tedious but routine to derive
>
(39)
But when every company's money wages go up, because of bigger b,
for b>b(0).
so do the prices of goods the workers buy,
and probably in about the same
proportion.
So far the situation has been fully analyzed in the unemployment region
b>b(0).
What happens, though, in the full-employment region b<b(0)?
The
methodology used to derive condition (23) is invalid under full employment
and some sort of a corner solution of V is implied, so that a different
approach is required.
Suppose, as will presently be confirmed, that when b<b(0) each firm
pays a competitive (full employment) wage W* and charges a correspondingly
marked-up price P* where
W*
^
=
^'^^X
(40)
^^°^
F(1)^(1)
^=4^
(41)
When L*=1 for each firm, condition (13) for
X->0
means that if any firm
pays a lower wage w<W*, it implies L=0 for that firm, whereas w>W* implies
L=L(w) satisfying (20),(2l).S
the competitive value W*,
Wow, should w be lowered in any firm to below
that would clearly decrease the firm's V- function,
since both pay and profits should then decline.
What about raising w above
W*, which would help the firm's insider workers'
pay but hurt its profits?
From (22),
16
Using definition (55), it is tedious but routine to verify that ex-
pression (42) is negative whenever b<b(0), so that the quasi-concave function
V(w) has a corner solution at w=W*.
Thus, a complete characterization of the equilibrium solution has
been derived for a wage economy.
Its main properties can be summarized as
follows.
Under the assumptions of the model, there exists a cutoff value b(0)
for the wage economy such that b<b(0) implies a competitive labor market
solution with u(b)=0, w(b)=W*, whereas b>b(0) implies a non-competitive labor
market solution with u(b)>0, w(b)>W*, and u'(b)>0, w'(b)>0.
The Profit Sharing Economy
Imagine now that the government has statutorily fixed X at some positive value. The long run equilibrium (15)-(17) can be solved as follows.
First suppose that
cj
has been negotiated at a sufficiently high value
that the profit sharing economy is in a regime with positive unemployment,
(it follows from the same logic as was developed in the pure wage case that,
independent of X, which just determines a multiplicative constant for the
profit function, there will be unemplojTaent if and only if
w
>
¥*
(43)
where the competitive wage ¥* is defined by (40).)
In such a situation,
each
firm is free to hire as much labor as it wants.
Then, suppressing extra independent variables where the meaning is
otherwise clear, each firm's profit function is
n(u);\)
=
(l-\)(R(F(L(co))) - a)L(w))
(44)
where
E'(F(L(a)))).F'(L(u))) =
c,
(45)
17
Each employed worker is then paid
(46)
and the Nashian arbitrator's maximand
=
V(a);\)
b log W(to;x)
+
(l-b) logUiuiiX)
(47)
will have the interior solution
V
^'- (1-b)^
=
=
(48)
Since from duality theory
=
n'(w;X)
-(l-x)L(aj)
(49)
the first order condition (48) can be rewritten as
1
=
S(L*)^E
(50)
where S(L) is defined by (33), and
E =
^^
(51)^
^-^
W
dco
is the elasticity of pay with respect to the base wage.
In order to evaluate E,
=
W(oj;\)
rewrite W, using (33), as
(l-\)a) + \aj(S(L(aj)) + l)
and differentiate (52) with respect to
V
co,
(52)
yielding
= ¥/u) + XajS'(L)L'(w)
(53)
Substituting from (53) into (51) yields
E =
1
+
>^u)
S'(L)L'(aj)
^^^>)
Since the second term in (54) is negative, because S'>0 and L'<0, it
can be deduced that
E
whenever \>0.
<
1
(55)
(Result (55), which is hardly surprising, merely states that
under profit sharing a 1% increase in base wages translates into a less than
18
^% increase in pay because of the adverse effects on gross profits.)
Using (55) and (34) to compare (50) with (31), (33), the following simple
qualitative comparison is immediate.
A profit-sharing economy has lower steady state unemployment than the
corresponding wage economy.
possible to make stronger quantitative statements if additional
It is
structure is put on the problem.
a
Suppose, as seems reasonable,
there exists
cutoff function b(\) such that
=> u
>
(or,
equivalently, a)>W*)
b(\) => u =
(or,
equivalently, a)<W*)
b > b(>^)
b
<
Then it can be shown that
b'(\)
0.
>
(56)
The proof is as follows. When b=b(x),
the equilibrium condition (50),
which is just holding on the borderline, becomes
1
S(1)^^^
=
E(x)
where E(\) is the value of (54) for L=1
E(x) =
+
1
(57)
,
aj=W*,
i.e.,
|(W*)2s'(l)L'(W*)
(58)
With (18) also holding in equilibrium,
(46)
becomes
W
=
W* + >,(M - W*)
(59)
^
=
!^
(60)
or
A.
+
M - W*
K
From (60) it follows that \/W is a monotonically increasing function of
E{x) in (58) is therefore declining in \
E'(^)
Combining
<
(61
0.
)
,
or
(61)
with (57) yields (56), the proposition to be proved.
19
When b>b(x), so that u>0 and co>W*, it can easily be shown that the
equilibrium price level is P
= co^/F'
=
M/F.
Thus far equilibrium has been analyzed for b>b(\), a situation of positive unemployment.
The rest of this section is devoted to characterizing an
equilibrium in the full employment case b<;b(x), u=0, a)<W*.
In the full employment case the equilibrium solution is
L* =
W* =
^
(62)
1
oj*
+ \(P*F(l)-co*)
r(1)
(63)
^^^^
TTT)
where W* is the competitive wage defined by (40).
Condition (63), which can be rewritten
w*i\) =
^(W*-XP*F(1))
is a basic theoretical result.
(65)
In any full employment equilibrium each work-
er is paid exactly the competitive wage W*.
in a full employment equilibrium,
When
X
is exogenously increased
the renegotiated base wage endogenously
declines according to formula (65) by just enough to maintain a perfectly
competitive pay level.
(in a situation where
oj*
<
W*
(66)
there are important implications for the disequilibrium behavior of the system.
When (66) holds, a profit-sharing economy with sticky pay parameters
(a)*A) remains at full employment even if M changes unexpectedly to any new
value in the range
n>i^^^011
.
When pay parameters are temporarily stuck (with quasi-fixed base wages
(67)
20
satisfying (66)) but all other variables are free to adjust, the short run
equilibrium price will accommodate according to the formula
P =
(68)
j^)
while hired labor and output remain at full employment levels.
By contrast,
in a wage economy, unexpectedly lower M in the presence of sticky wages
causes unemployment to go up.
The reason for these results is that each firm
in the long run equilibrium state of a profit-sharing economy, when (66)
holds, wishes to hire more labor on its existing contract but cannot find any
unemployed workers desiring to come on board.
This "excess demand for labor"
remains even after (small) changes in aggregate demand (coupled with sticky
pay parameters in the short run) temporarily force the system out of steady
state equilibrium, before pay parameters have had time to catch up.^)
The proof that (62)-(64) constitutes a long run equilibrium when b<b(x)
is as follows.
Let the economy be at full employment equilibrium L=1, w<W*,
P=M/F(1),
where the prevailing level of pay is
W =
^
+
\(PF(1) - w)
(69)
Given any parametrically fixed base wage
<
0)
let
L(co)
co
satisfying
¥*
from (11
(70)
),
(13) be the supply constrained solution of
Max R(F(L)) -
(c.
^(^^^^%I^^^))L
+
^
L
(71)
subject to:
,
If (70)
,(R(F(L))-c.L)
is in effect,
^
^
(^2)
the solution of (71), (72) must cause the con-
straint (72) to hold with full equality.
The reason is that if (72) has a
21
then (71), (72) is equivalent to the problem
zero shadow price,
Max (l->.)(R(F(L))-a)L)
(73)
L
which has the solution
E'(F(L))F'(L)
=
(74)
CO
implying
L >
whenever
L<
1
•
But
u)<W*.
Hence,
(75)
1
+
;,(
is a contradiction with the equilibrium condition
implies that (72) must hold with full equality, or
(70)
(.
(75)
R(F(L))-a)L ^ ^
h
Now when (76) holds,
of
co>
ing,
since
^
(^g)
the supply constrained firm's pay is independent
automatically adjusts, by long term attrition and new hir-
L(aj)
worker pay at W. In this case W=V independent of
to keep
co
,
and the
Nashian arbitrator's problem is reduced to maximizing the firm's profits
Max R(F(L(u3)))-(a)n ^^^^^iy^P~^^^'"^
With
L(cjj)
defined as the solution to
(71
)L(a))
)
(77)
under the constraint (76),
the optimization problem (77) can be rewritten simply as
Max (1-X)(R(F(L))-VL)
(78)
L
where the dummy variable w has been dropped because it is superfluous.
The solution of (78) is
R'(F(L*)).F'(L*)
=
W
(79)
From (79), then, there is full employment,
L* =
1
(80)
W*
(81
if and only if
W
=
)
22
Condition (63) is implied by (80),
(81),
and
Condition (64) is
(76).
immediate. This concludes the proof that (62)-(64) constitutes an equilibrium
solution in the full employment case.
Thus we have the following important observation.
Provided X is high
enough so that
b(0)
<
b < b(\)
(82)
insider bargaining power that creates long-run "natural" unemployment in a
wage economy will not cause unemployment in a profit-sharing economy.
A wage
system ends up in the excess supply of labor non-competitive wage region,
while a profit-sharing system will be in the competitive-pay excess demand
for labor region with correspondingly different consequences for macroeco-
nomic policy.
This result should be carefully understood.
upon a number of assumptions,
the beginning of the paper.
a
It
is contingent
the most important of which were outlined in
These assumptions seem quite reasonable to me as
generalized description of the world in which we live.
But they are likely
to be challenged by purists who tend to regard almost any violation of Pareto
optimality as prima facie ad hoc.
Suppose that in addition to (82) it is assumed that the markup of
prices over wages in a wage system is approximately independent of the em-
ployment level, or
^(L)/F'(L) ~ constant
(83)
Then the somewhat striking conclusion of this paper is that if workers were
forced to receive some part of their pay as a share of per-capita profits and
restricted to bargaining about base wages (or the opposite arrangement
—
fixed base wages and bargaining only about profit shares), everyone would be
better off.
be
Under the assumptions being made, real pay of the employed would
(approximately) the same in both systems.
But with profit sharing,
every
23
worker has a job, there is more output, real profits are higher, and prices
are lower for any given level of aggregate demand.
It is
important to understand where these results are coming from.
So
long as the finn determines employment to maximize profits given the pay
parameters, it will offset wage pushiness in a profit-sharing context by
hiring more outsider workers.
Since the insider workers cannot actually
obtain the higher pay they seek, and the only ultimate result of their wage
pushiness is to damage their firm's profitability, the weighted Nash bargaining solution becomes the competitive equilibrium.
down,
30 to speak,
on the side of the employer
The Nash arbitrator comes
(and,
without explicitly con-
sidering their needs, the outsider workers) because the employer has to give
up such a large amount of profit in order to improve the pay of the insider
workers even a slight amount.
Of course a collusive deal between firms and
insiders to restrict new hiring of outsiders could theoretically be made, in
which case, if collusion were taken to the extreme, wage and profit-sharing
economies would end up at the same long-run equilibrium.
But that is being
ruled out by the reasonable (to me) hypothesis that under capitalism the
employer basically determines the employment level.
Incomplete collusion to
restrict new hires would yield "in between" results that would soften, but
not
eliminate, the basic differences between wage and profit-sharing systems
being stressed in this paper.
The situation is very different in a wage economy.
There the insider
workers can directly negotiate the money wage they actually end up receiving
even after the employer makes the necessary adjustments at the expense of the
unemployed outsiders.
Note that the insiders of a wage economy do not actu-
ally end up better off when they all are playing the game of "push".
all insider v/orkers are pushing up money wages,
When
that just causes prices to
24
rise and the "natural rate" to go up without actually altering real wages.
Although profit sharing may result in a Pareto superior configuration,
it is not stable under free choice of payment mechanisms.
are objects of negotiation,
If both
oj
and X
the economy will end up with a wage system and
its associated steady state unemployment.
To see this clearly,
imagine a
profit-sharing system in its full-employment competitive-pay equilibrium.
The median worker would vote for conversion to an all-wage system at a some-
what higher than competitive wage level.
The firm would not like the idea,
and would react by laying off some workers, but with b large enough to satisfy (82)
the Nashian arbitrator would rule in favor of the insider workers.
When all firms and insider workers act this way, the profit-sharing system
unravels.
The insider workers do not actually gain anything from mass con-
version to a wage economy, because prices also go up, but self interest compels them to push in this direction.
desire to move toward profit sharing.
And from a wage economy they have no
A motion to introduce profit sharing
into a wage firm would be defeated by majority vote of the insiders,
since it
would result in the firm hiring outsider workers and lowering the pay of
insiders.
Of course if every firm converted to profit sharing there would be
no outsider unemployed workers to hire and the insiders would be no worse
off.
But in all instances the insiders will vote for pure wages over profit
shares to maintain their illusory prisoner's dilemma advantage.
Extensions and Conclusions
The main conclusion is fairly obvious.
Widespread profit sharing can
be a powerful aid to inflation-free full employment because it gives firms an
incentive to turn outsider unemployed workers into insiders.
But,
if this
model rings true, it is unlikely to evolve by itself because of a strong
25
public good or prisoner's dilemma aspect.
side effects of profit sharing,
ive role.
If society wants the beneficial
the government must play an actively support-
This role could take many forms.
Perhaps the most useful form of
support would be a substantial tax break for profit-sharing income
working person's capital gains tax"
—
to encourage firms and
—
"a
insider workers
to adopt high-X behavior that is publicly rational but privately irrational.
(Any preferential tax treatment of share income in specific instances should
be made conditional upon the firm and union explicitly agreeing that the firm
is able to hire as many outsider workers as it wants.)
that wage and profit-sharing systems yield about the same
It was noted
real pay to each employed worker provided (83) holds.
This is dependent upon
the coefficient of insider bargaining power b being the same for each firm.
Suppose,
that the coefficient of bargaining power for the insider
though,
workers of firm
i
is b-,
<
b.
which differs from firm to firm while always satis-
fying
b(0)
<
b(\)
(84)
Then the story will be very similar in the aggregate to the case b.=b.
Average real pay will be the same under wage and profit-sharing systems.
the insider worker from a high-b.
tem, whereas the low-b.
profit sharing.
firm will be better off under
wage sys-
insider worker will fare better under economy-wide
This does not at all mean that the low-b.
for profit sharing.
a
But
insider will vote
For reasons similar to those already explained in the
previous section, the low-b.
insider will vote against profit sharing for his
own firm even though he would personally get higher real pay if the entire
economy were converted to a profit-sharing system.
profit sharing were universal, the low-b-
Furthermore, even if
insider worker would vote to con-
vert to a wage although, when all insiders unravel the profit-sharing system
26
the low-b.
by so voting,
insider will be worse off.
Profit sharing can be a great equalizer of workers' incomes.
differing coefficients of bargaining strength b.,
a
Even with
profit-sharing system
will result in equal pay for equal work for all workers in all firms, and no
But a wage system will result in higher pay for the insiders
unemployment.
of a high-b.
firm than for the insiders of a low-b.
overall unemployment.
firm, as well as some
In a wage economy the Nashian arbitrator will award
wages (and, indirectly, unemployment) to a firm on the basis of whether its
insider workers enjoy greater or lesser bargaining strength.
But that same
Nashian arbitrator in the same situation will make equal pay parameter awards
to all workers in a profit sharing economy.
This is one reason why the
greatest resistance to profit sharing is likely to come from those sectors or
firms where powerful unions are able to extort above-competitive wages.
The
insiders in such situations will actually find their pay lowered relative to
others from conversion to profit sharing unless a government tax benefit or
sv.bsidy offsets the loss of their monopoly premium.
Of course all of the previous analysis ignores uncertainty.
Some
economists would argue that a profit-sharing system is inferior to a wage
system because it forces workers to bear some risk.
But such arguments typ-
ically embody a basic fallacy of composition that is related to confusion
about the insider-outsider distinction.
is more risky.
For insiders a profit sharing system
But a wage system is more risky for outsiders.
Actually,
when a complete analysis is performed which considers the situation not as
seen by an insider worker but by a neutral observer with a reasonably specified social welfare function defined over the entire population of insiders
and outsiders, it becomes abundantly clear that the welfare advantages of a
profit-sharing system are enormously greater than a wage system.
The basic
27
reason is not difficult to grasp.
A wage system allows huge first-order
Okun-gap losses of output and welfare to open up when
the national income pie evaporates with unemployment.
significant slice of
a
A
profit-sharing sys-
tem stabilizes aggregate output at the full employment level,
creating the
biggest possible national income pie, while permitting only small secondorder Harburger-triangle distortion losses to arise because some crumbs have
been randomly redistributed from one firm's workers to another's in what is
essentially a zero-sum game.^^
This paper has woven the insider-outsider paradigm into a theory of
wage-push inflation at any unemployment level below the "natural rate."
good is this story as an explanation of a positive NAIRU?
I
think the answer
depends, as it should, on particular circumstances of time and place.
most European economies today,
1
believe the model of this paper is
good description of the essential problem.
How
For the United States,
a
I
For
quite
am less
confident about this model capturing the predominant factor behind a positive
NAIRU, although
I
think it is correctly identifying an important element.
But in any case
I
would like to stress here that widespread profit sharing
could be a powerful tool for lowering the NAIRU even under alternative scenarios.
One popular interpretation of the "natural rate" of unemployment is
that it represents semi-permanent frictional unemployment, due to microeco-
nomic structural changes, which
cannot be reduced by pure macroeconomic
policies except temporarily and at the cost of increasing inflation.
wage system is heavily implicated in this concept of the NAIRU too.
But the
After
all, both wage and profit-sharing systems respond to shifts in relative de-
mand by sending a signal that eventually transfers workers out of a losing
firm or sector and over to a winner.
Under a wage system the signal to a
28
worker that his firm is a loser in the game of capitalist roulette, and it is
time to look for a new job with a winning firm,
is that the worker is laid
off and must suffer through an unemployment spell of some duration while
searching for the new job.
voluntarily lets go of
a
Under a profit-sharing system, no firm ever
worker because of weak demand.
It is
the worker who
chooses to leave because pay is too low relative to what is readily available
elsewhere at successful firms eager to include new workers into their current
profit-sharing payment plans.
There are yet other, more exotic explanations of the "natural rate,"
such as misperception theories, asymmetric information theories,
fairness
theories, efficiency wage hypotheses, hysteresis-like inertial theories, and
so forth. ^i
But in every case that
I
have examined, a profit-sharing system
would lower (or at least not raise) the steady state unemployment rate relative to a wage economy. ^2
Finally,
I
would like to relate the conclusions of this paper to the
predominant disequilibrium view of macroeconomics.
The mainstream explana-
tion of short-run unemployment is that it is caused by insufficient aggregate
demand (relative to sticky pay parameters).
But here again the unemployment
is inextricably tied up with a wage system.
When a sticky wage economy
resting at its (positive) natural rate is shocked by lower aggregate demand,
that causes additional unemployment.
But when a sticky profit-sharing
economy resting at its (near-zero) natural rate is subjected to an aggregate
demand shock it remains at full employment.
From all of these theoretical exercises considered together it seems
difficult not to draw the conclusion that a profit-sharing economy is more
likely to have lower unemployment than a wage economy.
29
Footnotes
1.
See Lindbeck and Snower [1984].
See also the supportive empirical
paper by Gregory [1985]2.
Aside from a great deal of casual empiricism, there is
al evidence for this proposition in a wage economy.
also some form-
See Oswald
[1984].
In a
wage firm where senior workers are essentially indifferent to unemployment,
the contract curve coincides with the labor demand curve,
so
that it is
efficient for insider workers and management to limit themselves to bargaining about the wage,
letting management decide the employment level.
It
must
be admitted that a distressingly large number of alternative hypotheses are
also available, with some formal evidence for them too.
for a survey.
I
See Farber
am therefore forced in this paper to rely on what
[
I
1985
person-
ally believe to be the best description of the hiring decision, yet without
being able to claim overwhelming theoretical or empirical evidence for it.
5.
This point was first noted by Leontief [1946].
See also McDonald and
Solow [1983].
e.g., Weitzman [1983],
4.
See,
5.
See Weitzman [1985].
6.
The basic points
I
[l985].
am trying to make would only be strengthened if the
number of firms were variable, with new firms entering or old firms exiting
in response to perceived profit or loss opportunities.
applies also to capital intensity.
The same comment
Both wage and profit-sharing systems
will, in equilibrium, hire capital to the point where the marginal revenue
product of capital equals the interest rate.
30
Rubinstein, and Wolinsky [1985]-
7.
See Binmore,
8.
Note that, strictly speaking, condition (13) is for a profit-sharing
economy.
A wage economy could either be
treated as the limit of a profit-
sharing economy as the profit-sharing coefficient goes to zero, or, equivalently,
(13)
could be modified in the obvious way to treat directly the
infinitely elastic supply of labor available to a firm in a full-employment
wage economy.
9-
These themes are developed much more fully in Weitzman [l985],
[l983],
q.v.
10.
The size of these firm specific redistribution losses is bound to be
further limited when it is considered that output and employment in the
profit-sharing macroeconomy as a whole is stabilized and private insurance of
profit shares could be used to make such losses even smaller.
11.
See Stiglitz [l984], and the references contained therein,
of some unemplojTnent theories.
for a survey
Gregory [1985] contains some interesting
ideas on why a system may adapt to a wide range of NAIEU values.
I
believe
these notions could be nicely integrated into the present framework.
12.
The one possible exception to lower equilibriiim unemployment under
profit sharing is the efficiency wage class of theories.
In that framework
wage and profit sharing systems could, depending on the specification, end up
at the same equilibrium, with profit sharing losing its "excess demand for
labor" property.
But even in the efficiency wage context,
the out-of-equi-
librium behavior of a" profit-sharing system, when pay parameters are sticky,
yields less unemployment than a wage system.
31
References
Binmore, K., Rubinstein, A., and Wolinsky, A.
Solution in Economic Modelling," L.S.E.
Discussion Paper 85/112.
"The Nash Bargaining
Theoretical Economics
[1985],
Farber, H. [1984], "The Analysis of Union Behavior," Forthcoming in Handbook
of Labor Economics, M.I.T. Working Paper No. 355.
Gregory, R. [l985], "Wages Policy and Unemployraent in Australia," Conference
on the Rise in Unemployment, Viay 1985-
Lindbeck, A., and Snower, D. [l984J, "Involuntary Unemployment as an Insider
Outsider Dilemma," Stockholm, Institute for International Economic
Studies Seminar Paper No.
309.
Leontief, W. [ 1946 J, "The Pure Theory of the Guaranteed Annual Wage Contract," Journal of Political Economy 54 (February), 76-79
McDonald, I. M., and Solow, R. M.
American Economic Review 71
[198I], "Wage Bargaining and Employment,"
(December), 896-908.
Oswald A. J. [1984], "Efficient Contracts Are on the Labor Demand Curve:
Theory and Facts," Princeton University Industrial Relations Section
Working Paper No. 178.
Shaked, A., and Sutton, J. [1983], "Involuntary Underemployment as a Perfect
Equilibium in a Bargaining Model," LSE Theoretical Economics Paper
83/63.
Stiglitz, J.E.
No. 1442
Tobin,
[1934],
"Theories of Wage Rigidity," NBER Working Paper
J. [1972], "Inflation and Unemployment," American Economic Review
LXII, 1, 1-18.
Weitzman, M.L. [l983], "Some Macroeconomic Implications of Alternative
Compensation Systems", Economic Journal (December).
Weitzman, M.L. [l985], "The Simple Macroeconomics of Profit Sharing,"
American Economic Review (December).
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